Algebra Master Cheat Sheet - IIM CAT 2026

1. Exponents and Logarithms

Logarithms and indices form the bedrock of algebraic manipulation in CAT. Questions frequently test base-changing tricks and non-linear system simplifications.

Core Rules of Exponents

For non-zero real numbers \(a\) and \(b\) and rational exponents \(m\) and \(n\):

• Product Rule: \(a^m \cdot a^n = a^{m+n}\)
• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{mn}\)
• Power of a Product: \((ab)^n = a^n b^n\)
• Negative Exponents: \(a^{-n} = \frac{1}{a^n}\)
• Radical Representation: \(a^{1/n} = \sqrt[n]{a}\)

Essential Logarithm Properties

For positive real numbers \(x, y\) and base \(b > 0\) (where \(b eq 1\)):

• Product Property: \(\log_b(xy) = \log_b x + \log_b y\)
• Quotient Property: \(\log_b\bigl(\frac{x}{y}\bigr) = \log_b x - \log_b y\)
• Power Property: \(\log_b(x^k) = k \log_b x\)
• Change of Base Formula: \(\log_b x = \frac{\log_a x}{\log_a b}\)

2. Polynomials and Factoring

Quadratic equations and high-degree polynomials are among the highest-yielding topics in the CAT Quantitative Aptitude section.

Special Products & Algebraic Identities

• Square Identities: \((a \pm b)^2 = a^2 \pm 2ab + b^2\)
• Difference of Squares: \(a^2 - b^2 = (a - b)(a + b)\)
• Sum and Difference of Cubes: \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)

Quadratic Equations & Root Analysis

For a general quadratic equation \(ax^2 + bx + c = 0\) with \(a eq 0\), the solutions are given by the Quadratic Formula:

The Discriminant, denoted as \(\Delta = b^2 - 4ac\), determines the nature of the roots:

• \(\Delta > 0\): Two distinct real roots.
• \(\Delta = 0\): One repeated real root (equal roots).
• \(\Delta < 0\): Two complex conjugate roots.

3. Sequences and Series

Questions on Progressions (AP, GP, HP, and AGP) test a candidate's ability to identify pattern structures and sum infinite series.

Arithmetic Sequence (AP)

Given first term \(a_1\) and common difference \(d\):

• General \(n\)-th Term: \(a_n = a_1 + (n-1)d\)
• Sum of First \(n\) Terms: \(S_n = \frac{n}{2}\bigl(2a_1 + (n-1)d\bigr) = \frac{n}{2}(a_1 + a_n)\)

Geometric Sequence (GP)

Given first term \(a_1\) and common ratio \(r\):

• General \(n\)-th Term: \(a_n = a_1 r^{n-1}\)
• Sum of First \(n\) Terms (for \(r eq 1\)): \(S_n = a_1 \frac{1-r^n}{1-r}\)
• Sum of Infinite GP (valid when \(|r| < 1\)): \(S_\infty = \frac{a_1}{1-r}\)

4. Complex Numbers

Although conceptual complex number applications in CAT are rare compared to standard equations, understanding basic forms provides a major edge in advanced algebra puzzles.

Let \(z = a + bi\), where \(i = \sqrt{-1}\) such that \(i^2 = -1\):

• Complex Conjugate: \(\bar{z} = a - bi\)
• Modulus/Magnitude: \(|z| = \sqrt{a^2 + b^2}\)
• Euler's Formula: \(e^{i\theta} = \cos\theta + i\sin\theta\)

5. Inequalities and Absolute Value

Modulus and inequalities are core components of CAT quantitative reasoning, frequently combined with functions and graphs.

• Bounded Modulus: \(|x| < c \iff -c < x < c\)
• Unbounded Modulus: \(|x| > c \iff x > c \text{ or } x < -c\)
• Triangle Inequality: \(|a + b| \le |a| + |b|\)

6. Binomial Theorem

The Binomial Theorem aids in evaluating large expansions, calculating remainders, and solving combinatorial summation problems.

Where the binomial coefficient \(\binom{n}{k}\) is computed as:

7. CAT Exam Pattern & Weightage Analysis

An analysis of recent IIM CAT papers shows that Algebra carries significant weightage in the Quantitative Aptitude section. Below is an overview of topic distribution and question difficulty.

8. Frequently Asked Questions (People Also Ask)

How important is Algebra for IIM CAT 2026?

Algebra is one of the most critical modules in CAT Quantitative Aptitude alongside Arithmetic. Expect 6 to 8 questions out of 22 QA questions directly testing algebraic concepts.

What are the crucial formulas to memorize in CAT Algebra?

Key formulas include the sum and product of roots of quadratic equations, logarithm base change properties, sum formulas for AP/GP, and standard modulus inequalities.

Where can I practice previous year CAT Algebra questions?

You can access and solve thousands of topic-wise previous year questions (PYQs) with detailed solutions on dedicated preparation portals and targeted study channels.